Determinant formulas relating to tableaux of bounded height

Chen et al. recently established bijections for (d+1)-noncrossing/nonnesting matchings, oscillating tableaux of bounded height d, and oscillating lattice walks in the d-dimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length n and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel's remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle.